Vibrational control of the reaction pathway in the H + CHD3 → H2 + CD3 reaction

An accurate full-dimensional quantum state-to-state simulation of the six-atom title reaction based on first-principles theory is reported. Counterintuitive effects are found: Increasing the energy in the reactant’s CD3 umbrella vibration reduces the energy in the corresponding product vibration. An in-depth analysis reveals the crucial role of the effective dynamical transition state: Its geometry is controlled by the vibrational states of the reactants and subsequently controls the quantum state distribution of the products. This finding enables generalizing the concept of transition state control of chemical reactions to the quantum state–specific level.

simulations of the reactant side that has been published elsewhere (35). Here, we present results obtained from three different basis set combinations: (  ,  ,  ), (  ,  ,  ) and (  ,  ,  ). We distinguish the sets via the name of the product basis (  ,  ,  or ). The time-independent basis sets used in all calculations are listed in Tab. S2. The reference temperature used in the thermal flux operator was T = 2000 K. The thermal flux eigenstates were propagated for 150 fs on the reactant side and 105 fs on the product side. The dividing surface in the transition-state region is placed at r = 120 a.u. while the dividing surface in the product asymptotic area was set to R = 500 a.u.. A quartic absorbing potential starting at r = 145 a.u. and a maximum strength of − ⋅ 10.368 eV was used to decouple the reactant channel. Another absorbing potential with the same maximum strength was placed in the reactant asymptotic region starting at R = 1585 a.u.
For the asymptotic projection, the first two vibrational states of H2 and the ground state and first five umbrella excitations of CD3 were calculated. The convergence of the states with respect to number of SPFs and time-independent basis functions was verified. The results on the product channel are accumulated over all relative rotations of H2 and CD3 that couple to vanishing total angular momentum.

Convergence
We compare results obtained with sets , and to test convergence of the results.
is our reference basis and and test convergence in the coordinates (r, s, t) and (R, S, T), respectively. Furthermore, we started independent calculations with additional SPFs in the remaining coordinates but stopped them after a few ten fs, because the additional SPFs remained unoccupied. Convergence of thermal flux eigenstates and real-time propagation to the reactant channel was confirmed in earlier studies (36,38). Fig. S2-S21 display state-to-state reaction probabilities for different vibrational and rotational states of the reactant, CHD3, and different product vibrational states of CD3 that are discussed in the main work. We find that results obtained with different basis sets match precisely for all vibrational and rotational states. Results typically differ at around 5% (and often less than that) which is smaller than the 10% error reported for reactant basis (35). Note that errors here include not only the error from the product side but also from thermal flux eigenstate calculation and real-time propagation to the reactant side for the reported basis sets.

Illustrative movies
Movies S1-S3 illustrate the calculations and results. Data corresponding to the density matrix obtained by averaging over all propagated wave packets (weighted according to the reference temperature of 2000 K) is displayed.
A trajectory-like view of the simulated reaction process is shown in Movie S1. Here the different atoms are located at the expectation values of coordinate values, i.e., at their mean position. The approach of the reactants is labeled using negative times. The positions corresponding to the initial wave packets employed in the numerical calculations, i.e., the thermal flux eigenstates, are reached at time t=0. The outgoing motion of the reactants is labeled using positive time.
Movie S2 illustrates the propagation in the reactant channel in more detail. 1D probability densities along the H-(H-CD3)-distance (R), the H-CD3-distance (r), and the umbrella bending angle are plotted. Probability densities in r and corresponding to the different relevant eigenstates of methane are also shown. They illustrate typical vibration amplitudes and help to rationalize the contribution of different vibrational excited states to the overall reactivity.
Movie S3 analogously illustrates the propagation in the product channel. Here a transformation of the coordinate system is required before the start of the propagation. 1D probability densities along the (H-H)-CD3-distance coordinate R', the H-H-distance coordinate r', and the umbrella bending angle are plotted. Probabilities densities in corresponding to the different relevant eigenstates of methyl are displayed for comparison. They help to rationalize the (averaged) product state distribution.

A Sudden Approximation Based Model
The accurately computed state-to-state reaction probabilities can be convincingly interpreted using the simple model which employs the sudden approximation and harmonic models for all vibrations. Thus, the C-H stretching motion in the CHD3 reactant is described by the Hamiltonian (atomic units and thus ℏ = 1 are used throughout this work) with the reduced mass 0 denotes the equilibrium distance, the vibrational frequency in harmonic approximation, and , , and denote the masses of the carbon, hydrogen, and deuterium atoms, respectively. Following the reduced-dimensional description developed by Palma and Clary (57), the motion in the umbrella bending vibration in the CHD3 reactant and CD3 product is described in harmonic approximation by the Hamiltonian where the reduced mass is defined by 1 = sin 2 0 ⋅ 3 3 + + cos 2 0 ⋅ 1 , 0 and are the equilibrium values of the umbrella bending angle and the C-D distance, and is the vibrational frequency in harmonic approximation. Note that the equilibrium values 0 and vibrational frequencies differ significantly for the CHD3 reactant and CD3 product.
Within the sudden approximation, transition probabilities are computed by directly mapping an initial wavefunction ϕ onto a set of final product states ψ ν , i.e., the transition probabilities w ν are given by the Franck-Condon factors = |⟨ϕ|ψ ν ⟩| 2 . The simplest model employs harmonic oscillator product states and an initial state given by a Gaussian wavefunction corresponding to the ground state of a displaced harmonic oscillator with same frequency. Then the transition probabilities for an oscillator with mass and frequency are given by where is the Huang-Rhys-Parameter and Δ the displacement of the center of relative to the equilibrium geometry of the harmonic oscillator. This simple model has been used to obtain the description presented in Fig. 2. The product state distributions presented in Fig.1 can be associated with Huang-Rhys parameters for the umbrella bending mode in the CD3 product. Reaction from the ground vibrational state of CHD3 shows a product distribution which peaks at a quantum number = 1 and shows approximately equal probabilites for the = 0 and = 2 product states. This product distribution corresponds to a Huang-Rhys parameter = √2. Reaction from the umbrella bending excited state of CHD3 ( (CHD 3 ) = 1) shows a product distribution with approximately equal probabilites for the = 0 and = 1 product states which corresponds to a Huang-Rhys-parameter = 1. Reaction from the C-H stretch excited state of CHD3 ( ℎ (CHD 3 ) = 1) shows a product distribution with approximately equal probabilites for the = 1 and = 2 product states corresponding to a Huang-Rhys parameter = 2. Based on these values of the Huang-Rhys parameter, the transition state wavepackets which give rise to these product state distributions can be reconstructed. Using = 444cm −1 , 0 = 90° , and                  Table S1. Single-particle function basis sets. Numbers of single-particle functions in the multilayer representation of the MCTDH wave functions used in real-time propagations on the product side.  Table S2. Time-independent basis set representations. Sizes and types of the time-independent (primitive) basis sets used to represent the bottom layer single-particle functions.

Movie Captions
Movie S1. Trajectory-like view of the simulated reaction process. The motion of the mean position of the atoms in course of the reaction process is illustrated. The approach of the reactants is labeled using negative times while the outgoing motion of the products is labeled by positive time (see text for more details).
Movie S2. Propagation in the reactant channel. 1D probability densities along the H-(H-CD3)distance (R), the H-CD3-distance (r), and the umbrella bending angle θ are shown. Probability densities in r and θ corresponding to different relevant eigenstates of methane indicate typical vibrational amplitudes (see text for more details).
Movie S3. Propagation in the product channel. 1D probability densities along the (H-H)-CD3distance (R'), the H-H-distance (r'), and the umbrella bending angle θ are shown. Probabilities densities in θ corresponding to different relevant eigenstates of methyl indicate typical vibrational amplitudes (see text for more details).